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Double Bruhat Cells and Total Positivity JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 12, Number 2, April 1999, Pages 335{380 S 0894-0347(99)00295-7 DOUBLE BRUHAT CELLS AND TOTAL POSITIVITY SERGEY FOMIN AND ANDREI ZELEVINSKY Contents 0. Introduction 336 1. Main results 337 1.1. Semisimple groups 337 1.2. Factorization problem 338 1.3. Total positivity 339 1.4. Generalized minors 339 1.5. The twist maps 340 1.6. Formulas for factorization parameters 341 1.7. Total positivity criteria 343 1.8. Fundamental determinantal identities 343 2. Preliminaries 344 2.1. Involutions 344 2.2. Commutation relations 345 2.3. Generalized determinantal identities 346 2.4. Affine coordinates in Schubert cells 349 2.5. y-coordinates in double Bruhat cells 351 2.6. Factorization problem in Schubert cells 353 2.7. Totally positive bases for N (w) 354 2.8. Total positivity in y-coordinates− 356 3. Proofs of the main results 357 3.1. Proofs of Theorems 1.1, 1.2, and 1.3 357 3.2. Proofs of Theorems 1.6 and 1.7 359 3.3. Proof of Theorem 1.9 360 3.4. Proofs of Theorems 1.11 and 1.12 363 4. GLn theory 364 4.1. Bruhat cells and double Bruhat cells for GLn 364 4.2. Factorization problem for GLn 365 4.3. The twist maps for GLn 367 4.4. Double pseudoline arrangements 369 4.5. Solution to the factorization problem 371 4.6. Applications to total positivity 374 References 379 Received by the editors February 12, 1998. 1991 Mathematics Subject Classification. Primary 22E46; Secondary 05E15, 15A23. Key words and phrases. Total positivity, semisimple groups, Bruhat decomposition. The authors were supported in part by NSF grants #DMS-9400914, #DMS-9625511, and #DMS-9700927, and by MSRI (NSF grant #DMS-9022140). c 1999 by Sergey Fomin and Andrei Zelevinsky 335 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 336 SERGEY FOMIN AND ANDREI ZELEVINSKY 0. Introduction This paper continues the algebraic study of total positivity in semisimple alge- braic groups undertaken in [3, 4]. Traditional theory of total positivity, pioneered in the 1930s by Gantmacher, Krein, and Schoenberg, studies matrices whose all mi- nors are nonnegative. Recently, G. Lusztig [18] extended this classical subject by introducing the totally nonnegative variety G 0 in an arbitrary reductive group G. (Lusztig’s study was motivated by surprising≥ connections he discovered between total positivity and his theory of canonical bases for quantum groups.) The main object of study in [3, 4] was the structure of the intersection G 0 N,whereNis a maximal unipotent subgroup in G. In this paper we extend≥ the∩ results of [3, 4] to the whole variety G 0. ≥ It turns out that the natural geometric framework for the study of G 0 is pro- vided by the decomposition of G into the disjoint union of double Bruhat≥ cells Gu,v = BuB B vB ;hereBand B are two opposite Borel subgroups in G, and u and v belong∩ − to− the Weyl group−W of G. We believe these double cells to be a very interesting object of study in their own right. The term “cells” might be misleading: in fact, the topology of Gu,v is in general quite nontrivial. (In some special cases, the “real part” of Gu,v was studied in [21, 22]. V. Deodhar [9] stud- ied the intersections BuB B vB whose properties are very different from those of Gu,v.) ∩ − We study a family of birational parametrizations of Gu,v, one for each reduced expression i of the element (u, v) in the Coxeter group W W .Everysuch parametrization can be thought of as a system of local coordinates× in Gu,v.We call these coordinates the factorization parameters associated to i. They are ob- tained by expressing a generic element x Gu,v as an element of the maximal torus H = B B multiplied by the product∈ of elements of various one-parameter subgroups in ∩G associated− with simple roots and their negatives; the reduced ex- pression i prescribes the order of factors in this product. The main technical result of this paper (Theorem 1.9) is an explicit formula for these factorization parameters as rational functions on the double Bruhat cell Gu,v. Theorem 1.9 is formulated in terms of a special family of regular functions ∆γ,δ on the group G. These functions are suitably normalized matrix coefficients cor- responding to pairs of extremal weights (γ,δ) in some fundamental representation of G. We believe these functions are very interesting subjects that merit further study. For the type A, they specialize to the minors of a matrix, and their proper- ties are of course developed in great detail. It would be very interesting to extend the main body of the classical theory of determinantal identities to the family of functions ∆γ,δ. In this paper, we make the first steps in this direction (see especially Theorems 1.16 and 1.17 below). As in [3, 4], the main algebraic relations involving factorization parameters and generalized minors ∆γ,δ can be written in a “subtraction-free” form, and thus the theory can be developed over an arbitrary semifield. The readers familiar with [3, 4] will have no trouble extending the corresponding results there (cf. [3, Section 2]) to the more general context of this paper. We do not pursue this path here since at the moment we have not developed applications of this more general setup. Returning to total positivity, our explicit formulas for factorization parameters allow us to obtain a family of total positivity criteria, each of which efficiently tests whether a given element x from an arbitrary double Bruhat cell Gu,v is totally License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DOUBLE BRUHAT CELLS AND TOTAL POSITIVITY 337 nonnegative. More specifically, each of our criteria consists in verifying whether x satisfies a system of inequalities of the form ∆γ,δ(x) > 0, the number of these inequalities being equal to the dimension of Gu,v (see Theorem 1.11). In Section 1 we give precise formulations of our main results. Their proofs are given in Sections 2 and 3. The last Section 4 contains applications of our theory to the case of the general linear group.ThecaseG=GLn is treated separately for a number of reasons. First, GLn is not a semisimple group (although everything reduces easily to SLn). Fur- thermore, the questions that we consider become some very natural linear-algebraic questions whose understanding does not require any Lie-theoretic background. For instance, the factorization parameters become the parameters in factorizations of a square matrix into the smallest possible number of elementary Jacobi matrices. Our main results seem to be new even in this case. In Section 4, we tried to present them in an elementary form, making this section as self-contained as possible. Last but not least, our results in the GLn case have a particularly transparent formulation in the language of pseudoline arrangements. The criteria of Theorem 1.11 lead us to new solutions of the classical problem of efficiently testing whether a given n n matrix is totally positive, i.e., has all minors > 0. This problem has a long× history. The ground was broken in 1912 by M. Fekete [10] who proved that positivity of all solid minors, i.e., those formed by several consecutive rows and several consecutive columns, is sufficient for total positivity. It took a while before it was realized that Fekete’s criterion was far from optimal: as shown in [13], it is enough to check the positivity of those solid minors that involve the first row or the first column of the matrix (this result can also be derived from [8]). Note that the number of such minors is n2 while the total 2n 2 number of minors is n 1. One can show that at least n minors are needed to characterize total positivity;− thus the criterion reproduced above is “minimal”. Theorem 4.13 (a specialization of Theorem 1.11) includes this criterion into a family of minimal total positivity criteria associated to shuffles of two reduced words for the permutation wo = nn 1 2 1. For example, for n = 3 we obtain 34 different criteria shown in Figure 8− at the··· end of the paper. 1. Main results 1.1. Semisimple groups. We begin by introducing general terminology and no- tation (mostly standard) for semisimple Lie groups and algebras (cf., e.g., [23]). Let g be a semisimple complex Lie algebra of rank r with the Cartan decomposition g = n h n.Letei,hi,fi ,fori=1,... ,r, be the standard generators of g, − ⊕ ⊕ and let A =(aij )betheCartan matrix.Thusaij = αj (hi), where α1,... ,αr h∗ are the simple roots of g.LetGbe a simply connected complex Lie group with∈ the Lie algebra g.LetN,Hand N be closed subgroups of G with Lie algebras n , h and n, respectively.− Thus H is a maximal torus,andNand N are two opposite− maximal unipotent subgroups of G.LetB =HN and B =− HN be − − the corresponding pair of opposite Borel subgroups.Fori=1,...,r and t C,we write ∈ (1.1) xi(t)=exp(tei) ,xi(t)=exp(tfi) , so that t xi(t)(resp.t xi(t)) is a one-parameter subgroup in N (resp. in N ). 7→ 7→ − We prefer the notation xi(t) to the usual yi(t), for reasons that will become clear later. It will be convenient to denote [1,r]= 1,..
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